# Tagged Questions

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### Height function on 2-torus with only 3 critical points

It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number ...
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### What is the analog of “monotonic” for scalar functions on surfaces?

"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...
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### Lusternik-Schnirelmann Theorem

In various paper i found this: But i don't find this theorem of Lusternik-Schnirelmann, have you an idea where i can find this theorem, the condition? Thank you.
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### Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...
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### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow \begin{cases} ‎\Delta^2u=‎‎\lambda ‎‎\dfrac{u}{|x|^4}‎‎+u^{‎p}‎ & \mathrm{in}‎\hspace{...
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### Set of critical values is compact [closed]

Let $M\subseteq \mathbb{R}^n$ be a compact manifold with $\partial M=\emptyset$ and let $f: M\rightarrow S^p$ be a smooth map. I want to unerstand the proof of the following lemma which occurs in ...
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### Continuity of critical points with respect to a parameterisation.

Hello. I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in ...
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### When is a critical value of a map contained in the interior of the image?

Let $M^n$ be a compact manifold, and $F\colon M \to \mathbb{R}^n$ a smooth map. The inverse function theorem implies that every regular value of $F$ lies in the interior of $F(M)$, hence every point ...
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### Lower bound on the diameter of a ball contained in the stable manifold of a critical point

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Consider the negative of the gradient ...
I have a question about a simple proposition, I suppose that this is something well-known or a special case of something well-known: Let $D\subset\mathbb{R}^{2}$ be the closed unit disk in the plane ...
I have a multivariate polynomial function of N variables $f(x_1,x_2,…,x_N) = x_1 x_2 x_3 .. x_N \left( 1 + \sum_i^N (a_i x_i^2 - x_i) \right)$, where $a_i > 0$ are real positive numbers. By ...